What question did this study set out to answer?

The study aims to explore and characterize the tail medians of normalized prime gaps at specific threshold levels.

April 18, 2026Open Access

A Structured Plateau in the Tail Medians of Normalized Prime Gaps

Key Points

The study aims to explore and characterize the tail medians of normalized prime gaps at specific threshold levels.
Utilized a deterministic bucket-pack pipeline for data processing.
Analyzed prime gaps observed up to 10^13.
Examined threshold levels at z ∈ {2.5, 2.6, 2.7, 2.8, 2.9, 3.0}.
Focused on the (L,z)-surface and its bounded range properties.
Found that the (L,z)-surface remains within a bounded range across all thresholds.
Observed a visible internal structure in the tail medians at the tested thresholds.
Established an empirical surface that can be used for further analysis.

Abstract

We study the thresholded tail medians of normalized prime gaps through the plateau-height coordinate ΔL (z) = mL (z) - z, where mL (z) is the exact median of Xₙ = gₙ/log qₙ conditional on Xₙ ≥ z. Using a deterministic bucket-pack pipeline over the observed range up to 10¹3, we analyze the main threshold set z ∈ 2. 5, 2. 6, 2. 7, 2. 8, 2. 9, 3. 0. Across all six thresholds, the observed (L, z) -surface stays within a bounded range and exhibits visible internal structure. The result is observational: it fixes a reproducible empirical surface together with supporting baseline and local-summary context, and it provides a calibration target for later explanatory or comparative work.

A Structured Plateau in the Tail Medians of Normalized Prime Gaps

Key Points

Abstract

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